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6

Nomarski imaging interferometry to measure

the displacement field of MEMS

Fabien Amiot, Jean Paul Roger

November 1, 2018

Abstract

We propose to use a Nomarski imaging interferometer to measure

the out-of-plane displacement field of MEMS. It is shown that the

measured optical phase arises both from height and slope gradients.

Using four integrating buckets a more efficient approach to unwrap

the measured phase is presented, thus making the method well suited

for highly curved objects. Slope and height effects are then decou-

pled by expanding the displacement field on a functions basis, and

the inverse transformation is applied to get a displacement field from

a measure of the optical phase map change with a mechanical load-

ing. A measurement reproducibility of about 10 pm is achieved, and

typical results are shown on a microcantilever under thermal actua-

tion, thereby proving the ability of such a set-up to provide a reliable

full-field kinematic measurement without surface modification.

1 Introduction

The increasing interest for micro-electro-mechanical systems (MEMS), es-pecially when they are used as micro-mechanical sensors [1], leads one tofocus on the mechanical behavior of micro-objects. First, the standardizedmechanical tests at the macro scale have been adapted to the micro one, as-suming an homogeneous stress or strain state. Bending [2] and tensile tests[3, 4], as well as fatigue or creep tests [5] are performed since years, providinga global kinematic response of the tested object.

However, classical photolithography processes use visible light to transfera mask onto a wafer surface. Then, light diffraction limits the achievable

1

accuracy of the geometric shape of the resulting micro-objects, and the di-mensions margins tends to increase compared to the dimensions themselvesas the object’s size decreases. The homogeneous stress (or strain) assump-tion usually satisfied when performing mechanical tests at the macro-scaleis no longer reasonable, and one has to deal with heterogeneous mechanicaltests. As a consequence, one has to perform a spatially resolved kinematicmeasurement instead of a global one.

Moreover, as their size decreases, the surface on volume ratio significantlyincreases, and the behavior of micro-objects tends to be dominated by surfaceeffects instead of volume ones. As a consequence, measuring the displacementfield should avoid any surface modification or any contact. In addition, thesurface roughness of MEMS is usually very low, so that measurement tech-niques involving surface-generated speckle [6] may be difficult to implement[7].

An optical interferometric imaging set-up is then well suited to measuredisplacement fields. The polarization interferometer [8] proposed herein isderived from the one initially proposed by Nomarski [9]. Dealing with surfacetopographies, it has been previously used to determine the mean slope oftilted samples [10] or to get an image of the roughness of polished surfacesusing a multichannel Nomarski microscope [11, 12, 13].

After recalling the relation between the optical phase induced by thesample-Wollaston prism group and the measured intensity, the twofold originof the optical phase is detailed. A shot-noise limited detection is described,and the inversion method developed to convert an optical phase change intoan out-of-plane displacement is presented. An example is finally providedon the measurement of the displacement field of a thermally loaded micro-cantilever.

2 Measuring a differential topography

2.1 Experimental set-up

The basic interferential microscopy imaging set-up is shown in Fig. 1. Aspatially incoherent light source (LED, λ = 760 nm) illuminates a polariza-tion beam-splitter. The beam reflected by the beam-splitter goes through apolarization modulator and is initially polarized at 45˚ of the axes of a Wol-laston prism. This splits the beam into two orthogonally polarized beams at

2

a small angle between each other. These beams are focused upon the sampleby an objective lens. After reflection and recombination by the Wollastonprism, the beam goes through the polarization modulator and the polariza-tion beam-splitter. The transmitted beam is finally focused on a CCD array.The polarization beam-splitter behaves as crossed linear polarizers mountedat 45˚ of the axes of the Wollaston prism and of the polarization modulator.One finally gets the interference between two images of the sample, shiftedby the Wollaston prism of the distance d. The resulting interference patternis recorded on a CCD array (DALSA CA-D1, 256× 256 pixels, 8 bits).

2.2 Interference pattern obtained with a Nomarski imag-

ing interferometer

Let us denote p (Fig.1) the polarisation direction of the beam incident onthe Wollaston prism. The Wollaston prism shear-direction is denoted y. Thevector a denotes the polarisation direction of the beam impinging on the CCDarray. The orthogonal directions, in the prism’s plane, are denoted q, x, b,respectively. One uses a light emitting diode, so that we denote E0 the am-plitude of the non polarized wave impinging on the polarizing beam-splitter.If tp is its amplitude transmission factor, and ǫp the attenuation factor forthe (ideally) suppressed component, the Wollaston prism is illuminated bytwo orthogonally polarized beams, which electric fields are Eppp and Epqq,with

Epp = E0tp (1)

Epq = E0ǫptp (2)

For a non polarized light source (LED), these two beams are incoherent,and should be treated separately. ǫptp is the transmission factor in the stopdirection, so that ǫp = 0 is the perfect polarizer case. For each beam, theWollaston prism splits the beam into two orthogonally polarized beams (i.e.,a x and a y component), and the light goes through the path (objective- sample - objective - Wollaston prism). Let us consider that this resultsin a phase shift φ between the x and the y components. Let us denote tnthe transmission factor of the path (Wollaston prism - objective - sample -objective - Wollaston prism), ta the analyser’s transmission factor, and ǫataits further attenuation factor in the stop direction. The total intensity Iimpinging locally on the CCD array is the sum of the intensity arising from

3

the two incoherent beamsI = Ip + Iq (3)

Assuming that (x,p) = (x,b) = π4, one obtains [10]

I = I0 + A cos(φ) (4)

I0 =|E0tatntp|2

2(1 + ǫ2a)(1 + ǫ2p) (5)

A =|E0tatntp|2

2(1− ǫ2a)(1− ǫ2p) (6)

The contrast of the interference pattern

A

I0=

(1− ǫ2a)(1− ǫ2p)

(1 + ǫ2a)(1 + ǫ2p)(7)

equals 1 for perfect polarizers (ǫa = ǫp = 0), and decreases when ǫa orǫp increase. Adding an error on the orientation of the polarization beam-splitter leads to the same expression, so that the equation (4) is consideredgeneral enough to describe real interference patterns. A typical interferencepattern obtained in water with two 70 micrometers long micro-cantileversand a shear distance d ≃ 50µm is shown on figure 2. The two shearedimages are clearly distinguishable. The optical phase range covers almost 15interference fringes. The closely packed interference fringes, as well as thequite short correlation length of the used light source (almost 15µm) reducethe contrast of the interference pattern, thereby limiting the quantitative useof obtained phase map. Let us assume that the optical phase φ introducedby the path (Wollaston prism - objective - sample - objective - Wollastonprism) may be decomposed in a term φ0 arising from the Wollaston prismand a contribution φm arising from the object

φ = φ0 + φm (8)

The section 2.3 exhibits the phase directly arising from the Wollaston prism,whereas the section 2.4 exhibits the phase arising from the topography of thesample.

2.3 Optical phase arising from the Wollaston prism

2.3.1 Optical path functions

Let us consider for simplicity an “ideal” Wollaston prism, which geometry isdescribed in the plane defined by both the optical axis of the system and the

4

y direction on figure 3. Fig. 3 shows the decomposition of a ray impingingorthogonally on the prism into two emerging rays and the angles definition.Snell-Descartes’ laws at the interface between the two half-prisms read

nE sin(βTM) = no sin(θ) (9)

no sin(βTE) = nE sin(θ) (10)

where no is the ordinary refractive index of the used birefringent material, andnE is the extraordinary one. Assuming an ambiant media with a refractiveindex equals to 1, Snell-Descartes’ laws for the exit face of the prism read

sin(βTMair ) = nE sin(βTM − θ) (11)

sin(βTEair ) = no sin(βTE − θ) (12)

If all the angles are small

βTMair − βTEair ≃ 2(no − nE)θ (13)

The emerging rays appear to split on a plane, called plane of apparent split-ting (PAS), which is here located inside the prism. The use of a modifiedWollaston prism [14] allows one to move the PAS outside the prism. Thisplane is considered to be perpendicular to the figure’s plane. Its position isone of the fundamental characteristics of the prism. Investigating the imag-ing properties of the system, let us denote α1 the angle of an impinging raywith respect to the normal of the entrance interface in the figure’s plane.Considering any dependance on α1 is then moving in the field of view alongthe shear direction. The optical path travelled by the TE (resp. TM) polar-ized ray through the prism lTE(y, α1) (resp. l

TM(y, α1)) when the apparentsplitting occurs at position y on the PAS (the origin will be defined later)depends on α1. Assuming that all the angles are small,

lTE(y, α1) ≃ nE

(

h

2+ yθ

)

+ no

(

h

2− yθ

)

+hα1n1θ

2

(

nonE

+nEno

)

lTM(y, α1) ≃ nE

(

h

2− yθ

)

+ no

(

h

2+ yθ

)

+hα1n1θ

2

(

nEno

(

2− nEno

)

+ 1

)

(14)

These two functions will be referred as optical path functions.

5

2.3.2 Optical phase when the PAS matches the objective rear

focal plane

Assuming that the PAS of the prism matches the objective rear focal plane,Fig. 4 presents the ray tracing for the two emerging rays of Fig. 3. These twoemerging rays intersecting the PAS at the ypas cross the PAS, after reflectionon a plane object, at the the same point yR,TE = yR,TM when the object isorthogonal to the lens axis. This last point is symmetric of the first withrespect to the lens axis. Let us thus define the median plane of the prism πmas orthogonal to both the figure’s plane and the PAS, passing by the O point(see Fig. 3). This point is defined to satisfy the condition EO = OS. Letus then consider that the distance between the πm plane and the symmetryaxis of the lens is Tw. Any point of the PAS may be described either bythe abscissa y in the prism’s frame (the origin is at the intersection withπm) or by the abscissa y = y − Tw (relative to the symmetry axis of thelens). In the prism’s frame, the rays emerge from the PAS at ypas = Tw − yt,and the reflected rays go through the PAS at yR,TE = yR,TM = Tw + yt. Theoptical paths lTEar and lTMar travelled to the sample and back are deduced fromlTE(y, α1) and l

TM(y, α1) and allow one to compute the optical phase

φW0 =2π

λ(lTEar −lTMar ) =

2π

λθ

(

4(nE − no)Tw + hα1n1

(

nonE

+nEno

(nEno

− 1)− 1

))

(15)The first term in (15) no longer depends on yt but on the position of the prismwith respect to the symmetry axis of the lens Tw, and is homogeneous in thefield of view. If the birefringence of the used material is denoted nE − no =noǫ, the additional optical phase difference (second term) proportional to theincidence angle α1 is shown to grow as ǫ2 whereas the first term scales asǫ, so that the added optical phase φW0 may be considered homogeneous inthe field of view. For a quartz-made Wollaston, ǫ ≃ 10−2 so that the addedoptical phase may be considered homogeneous in the field of view. Thisis no longer true when the PAS doesn’t match the rear focal plane of theobjective, because the reflected rays no longer cross the PAS at the samepoint (see Fig. 4). Assuming that the PAS remains parallel to the rear focalplane, at a distance δPAS and using the same method, the additional opticalphase difference φPAS is

φPAS ≃ 8π

λθα1δPAS(nE − no) (16)

6

thus introducing a linear phase along the Wollaston shear direction.

2.4 Optical phase arising from the object topography

2.4.1 Optical phase arising from height variations

The previous ray tracing is shown in the case of a stepped sample (height∆z) in Fig. 5. Thanks to the Fermat’s principle, tilting the sample doesn’tinduce an extra phase shift in the objective-sample path (i.e., regardless ofthe phase shift induced by the Wollaston prism). The dot line is for theray reflected under the previous conditions. The optical phase difference φharising from the step corresponds to a travel along [AB] and [BC], whereC is the orthogonal projection of A on [BD]. In a medium which refractiveindex is n, this optical phase difference reads

φh =2π

λn× (AB +BC) =

4π

λn∆z cos(α) (17)

where α is the incidence angle on the object. If the numerical aperture of theobjective is low enough, α remains small and the above expression expands

φh ≃4πn∆z

λ(18)

As the numerical aperture of the objective increases, the previous expansionis no longer valid and the equation (18) is replaced, for numerical aperturesless than 0.3, by

φh =4πn∆z

ιλ(19)

where ι is a scale parameter depending on the numerical aperture (see ap-pendix A).

2.4.2 Optical phase arising from the local surface orientation

In section D2.4.1, the two orthogonally polarized rays cross on the PAS. Thisis no longer true if the two rays experience a different surface orientation.Let us denote by γTE (resp. γTM) the local surface orientation experiencedby the TE (resp. TM) ray. For a ray emerging from the PAS at the positiony (in the prism’s frame), the reflected rays cross the PAS at the abscissa

7

(figure 6)

yR,TE =−y + f tan(2γTE)

1 + y

ftan(2γTE)

(20)

yR,TM =−y + f tan(2γTM)

1 + y

ftan(2γTM)

(21)

Where f is the focal length of the objective lens. The optical path travelledby the two rays reads (with α1 = 0)

lTEar ≃ lTE(Tw + y, 0) + lTE(Tw + yR,TE, 0) (22)

lTMar ≃ lTM(Tw + y, 0) + lTM (Tw + yR,TM , 0) (23)

and the optical phase difference reads (to the first order with respect to thesurface orientation)

φW =2π

λ(lTEar −lTMar ) =

4π

λ(nE−no) tan(θ)

(

1 +

(

y

f

)2)

f(γTE+γTM) (24)

One should highlight that this optical phase difference doesn’t arise from thesample itself, but from the optical phase imbalance it introduces when therays travel back through the Wollaston prism. Moreover, this phase differencedepends on the abscissa y, and then on the angle α. It’s also worth notingthat a global tilt of the specimen (i.e., γTE = γTM 6= 0) induces an extraoptical phase. It arises from the previous remarks that the relation betweenthe local surface orientations and the induced optical phase depends on theprism-objective group. This relation may be rewritten

φW = φori + φtilt =∂φW∂γ

(γTE − γTM) + φtilt (25)

with

φtilt = 2∂φW∂γ

γTM (26)

The scalar ∂φW∂γ

then describes the prism-objective group. It may be obtainedwhen measuring the optical phase when tilting a reasonably flat mirror. Thefigure 7 shows the result of such a calibration for a quartz Wollaston prism(θ ≃ 18) and a water immersion objective lens (focal length 18mm, NA 0.3,

8

used to obtain the interference pattern in Fig.2). The relation is linear, andthe fitted slope is

2∂φW∂γ

= 1.1× 103 (27)

This simple experiment allows one to obtain the sensibility ∂φW∂γ

to a localorientation gap by tilting the whole sample.

Finally, the total measured optical phase difference φ may be written asthe following sum

φ = φW0(Tw) + φPAS(δPAS) + φh(∆z) + φW (γTE, γTM) (28)

where the two first terms (related to φ0) account for the path differenceintroduced by the Wollaston prism itself : an homogeneous term φW0 anda linear term along the shear direction φPAS which vanishes when the PASmatches the rear focal plane of the objective lens. The two last field terms(related to φm) originate from the surface topography : height variations φhand slope field, φW including the contribution of the mean tilt of the sampleφtilt as well as slope variations φori.

3 Phase integration, retrieval and unwrap-

ping

3.1 Principle

The optical flux collected by the pixel indexed by (l, m) on the CCD matrixcan be formally written according to (4) as

I(l, m, t) = I0 + A cos[φ(l, m) + ψmod(t)], (29)

where φ(l, m), which is to be determined, is the optical phase introduced bythe sample and the Wollaston prism. The phase modulation introduced bythe photoelastic modulator reads

ψmod(t) = ψ0 sin(2πfmodt+ θmod). (30)

The angles ψ0 et θmod are two parameters that can be chosen among manycouples. The algorithm to obtain φ uses four integrating buckets [15]: if T = 1/fmod

9

is the modulation period, four images of the interference pattern can be cap-tured during the period T , so that each image results from the integrationof the optical flux during a quarter of one period. One obtains four imagesEp, for p = 1, 2, 3, 4

Ep =

∫pT

4

(p−1)T4

I(t)dt (31)

Combining with Eq. (29) allows one to obtain

Ep =T

4(I0 + AJ0(ψ) cos(φ))

+TA cos(φ)

π

∞∑

n=1

J2n(ψ0)

2n[sin(npπ + 2nθmod)− sin(n(p− 1)π + 2nθmod)]

− TA sin(φ)

π

∞∑

n=0

J2n+1(ψ0)

2n+ 1

×(

cos(π

2(2n+ 1)(p− 1) + (2n + 1)θmod)

− cos(π

2(2n+ 1)p+ (2n+ 1)θmod)

)

(32)

where Jn is the first kind Bessel function of n order. The images Ep dependon I0, cos(φ) and sin(φ), so that using four independent images providesenough information to recover φ. Classical algorithms [15] use particularlinear combinations

Σs = −(E1 − E2 −E3 + E4) =4TAπ

Γs sin(φ)Σc = −(E1 − E2 + E3 −E4) =

4TAπ

Γc cos(φ)(33)

withΓs =

∑

∞

n=0(−1)n J2n+1(ψ0)2n+1

sin [(2n+ 1)θmod]

Γc =∑

∞

n=0J4n+2(ψ0)

2n+1sin [2(2n+ 1)θmod]

(34)

The optical phase is then usually recovered by using an “atan2” functionwith the arguments provided by the equations (33) with a (ψ0, θmod) couplesatisfying Γs = Γc. The indicator

Υ2 = Σ2c + Σ2

s (35)

is then independent of the phase φ and defines the intensity of the signalexperiencing the phase φ. As a consequence, the phase obtained from the

10

previous algorithm is reliable provided that Υ2 is high enough, i.e., if thefringe contrast A

I0is sufficient. This condition may be difficult to satisfy when

the topography is described by a large number of closely packed interferencefringes (see Fig.2). It’s worth noting that the set of four images provides anover-determinated set of data. This redundancy is then exploited to providea more reliable phase measurement.

3.2 Least-square phase retrieval

For each pixel of the CCD array, the set of Eq. (32) may we rewritten as alinear system of equations

MP = E (36)

where the parameters vector P reads ((·)t denotes the transpose of (·))

Pt =

[

TI04,TA

πcos(φ),

TA

πsin(φ)

]

(37)

and the images vectorEt = [E1, E2, E3, E4] (38)

The matrix M is built from the modulation parameters

M =

1 c(1, ψ0, θmod) s(1, ψ0, θmod)1 c(2, ψ0, θmod) s(2, ψ0, θmod)1 c(3, ψ0, θmod) s(3, ψ0, θmod)1 c(4, ψ0, θmod) s(4, ψ0, θmod)

(39)

with

c(p, ψ0, θmod) =∞∑

n=1

J2n(ψ0)

2n[sin(npπ + 2nθmod)− sin(n(p− 1)π + 2nθmod)](40)

s(p, ψ0, θmod) = −∞∑

n=0

J2n+1(ψ0)

2n+ 1

×(

cos(π

2(2n+ 1)(p− 1) + (2n + 1)θmod)

− cos(π

2(2n+ 1)p+ (2n+ 1)θmod)

)

(41)

11

The matrix M is then independent of the considered point. For each pixelof the CCD array, the four images describe the vector E, and the solutionparameters vector Psol is obtained as a minimizer of

η2(P) = (MP−E)t (MP−E) (42)

and is the solution of the square linear system

MtMPsol = MtE (43)

The couple(

TA cos(φsol)π

, TA sin(φsol)π

)

is then extracted, and used as the argu-

ment of a standard “atan2” function to provide a less corrupted value of thephase. The figure 8 shows the phase map obtained from the scene presentedin Fig. 2 using the least-square algorithm.

3.3 Phase unwrapping

The use of the phase integration technique also allows one to reconsider thephase unwrapping problem. In the previous section, the information derivedfrom the experiments is a couple (X, Y ) proportional to (cos(φ), sin(φ))

(X, Y ) = C(cos(φ), sin(φ)) (44)

where C = 4TAπ

Γs,c when the linear combinations (33) are used, and C = TAπ

when the least-square algorithm is used. In the first case, the couples (X, Y )and (Σc,Σs) are equal. For a sake of simplicity, let us consider that a wrappedvalue of the phase φp is then obtained by using

φp = atan2(Y,X) (45)

φp lies then in [−π, π]. Recovering the unwrapped value of the phase mayturn into a brain-racking task when the phase-map is poorly discretized andthe phase jumps are densely packed. Numerous algorithms are available inthe literature, based either on local phase unwrapping [16] or on global phaseunwrapping [17]. Contrary to most of these algorithms, which use a wrappedphase map as an input, we propose here to use the two fields X and Y .

The figure 9 shows a typical case of phase jump between two adjacentpoints 1 and 2. Their true phase φ1 and φ2 is under scrutiny. AsX1 ≃ X2 6= 0and Y1Y2 < 0, using the “atan2” function induces a phase jump when moving

12

from 1 (φp,1 ≃ π) to 2 (φp,2 ≃ −π). Let us then define the frame (−→Xb,

−→Y b),

obtained by rotating the (−→X,

−→Y ) frame until the point 1 (X1, Y1) belongs to

the (Ω,−→Xb) axis. The phase φ2 − φ1 of the point 2 in the frame (

−→Xb,

−→Y b) is

then obtained according to

tan[φ2 − φ1] =−X2sin(φ1) + Y2cos(φ1)

X2cos(φ1) + Y2sin(φ1)(46)

providing the gap between the true phase values φ1 and φ2, thus defining(if 1 and 2 are for adjacent pixels), the true phase gradient, modulo 2π.The couple (X2, Y2) is obtained of the four images using one of the twoalgorithms previously described. cos(φ1) and sin(φ1) are deduced from thewrapped value φp,1. The phase gradient may then be integrated to providea true phase map. The figure 10 shows the phase map obtained from thewrapped phase map shown in Fig.8 using the described phase unwrappingtechnique.

3.4 Reproducibility of the phase measurement

The phase measurement reproducibility is assessed by measuring twice thephase map arising from the same differential topography of a reflective object.One gets two phase fields φ−(l, m) and φ+(l, m). Assuming that each of thesefields is the sum of deterministic part φd(l, m) and of a random part

φ−(l, m) = φd(l, m) + b−(l, m) (47)

φ+(l, m) = φd(l, m) + b+(l, m) (48)

the difference between the two phase fields provides then a realization of thedifference between two realizations of the noise b(l, m)

(φ+ − φ−)(l, m) = (b+ − b−)(l, m) (49)

The figure 11 shows a typical probability density of the variable (φ+ −φ−) obtained with the set-up described in Section 2.1. The dots are theexperimental distribution, whereas the solid line is a least-square fit using aGaussian, zero-mean, distribution. The agreement is excellent, and allow usto consider (φ+−φ−) as a random real number of variance 2σ2, if b is a randomvariable which variance is σ2. The figure 12 shows the evolution of

√2σ2

(converted to heights variation assuming (18)) as a function of the exposure

13

time texp used to form each intensity image. The experimental variance

grows as t−

12

exp, until the exposure time reaches several tens of seconds. Thereproducibility is therefore controlled by the exposure time, and the achievedlevel is (converted to height variations) close to 10 pm.

4 Recovering the displacement field

4.1 Principle

The calculations presented in sections 22.3 and 22.4 allow one to derive anexpression for the measured phase φ as a function of the surface topography.Assuming that the pixel size (in the object plane) is small enough comparingto the object’s size, one proposes for a sake of simplicity to formulate theinversion problem in the object’s plane. Inserting Eq. (19) and Eq. (25) intoEq. (28) allows one to write the total measured phase difference at the point(x, y)

φ(x, y) = φ0(x, y) + φtilt + Φ(x, y) (50)

Assuming that the shear direction is parallel to the y axis, and that thelocal surface orientation γ in Eq. (25) is given by the first derivative ofthe topography, the relation between the measured information Φ and thetopography reads, for a shear distance d

Φ(x, y) =4πn

ιλ

(

z(x, y +d

2)− z(x, y − d

2)

)

+∂φW∂γ

(

∂z

∂y(x, y +

d

2)− ∂z

∂y(x, y − d

2)

)

(51)where z(x, y) is the topography. To discriminate between the phase aris-ing from the slope variations and those arising from height variations, oneproposes to expand the topography on a functions basis

z(x, y) =∑

s

µszs(x, y) (52)

so that the reference (initial) measured phase field reads

φref(x, y) = φ0(x, y) + φtilt +∑

s

µsΦs(x, y) (53)

One should highlight that the shearing amount d is here introduced explicitlyto compute the functions basis Φs(x, y) by inserting Eq.(52) into Eq.(51). If

14

the topography is subjected to an out-of-plane displacement field w(x), φrefis changed into φw and the new topography is described by

z(x, y) + w(x, y) =∑

s

(µs + νs)zs(x, y) (54)

so that the displacement field is also expanded on the same functions basis

w(x, y) =∑

s

νszs(x, y) (55)

The knowledge of the numerical coefficients involved in the definition (51) ofΦ(x, y) combined with the choice of a functions basis allows one to recoverthe displacement field by computing the νs coefficients minimizing

η2ν(ν) =

∫

(

∑

s

νsΦs(x, y)− (φw − φref)(x, y)

)2

dxdy (56)

that is solving the linear system

N ν = F (57)

with

Nsi =

∫

Φs(x, y)Φi(x, y)dxdy (58)

Fs =

∫

Φs(x, y)(φw − φref)dxdy (59)

4.2 Example

The micro-cantilevers shown on Fig.2 are placed into a fluid cell, filled withmilliQ water which temperature is controlled thanks to a feedback loop con-trolled Peltier device. These cantilevers are multi-layer cantilevers, madeof a silica layer (770 nm), a titanium layer (20 nm) and a gold layer (50nm). These cantilevers are then subjected to a bimaterial effect, since thecoefficient of thermal expansion of the gold layer is almost ten the one ofthe silica layer. The optical set-up is also used with the Wollaston prism-objective couple calibrated in Section 2.4.2. A reference phase map of thecantilever is captured at 22.6˚C. The temperature is then increased to reach

15

24.1˚C, and a new phase map is captured. The phase gradients are com-puted according to the algorithm described in Section 33.3. Recovering thetwo-dimensional phase map from the phase gradients is an over-constrainedproblem, so that it is possible to avoid some unreliable data. Reliable dataare then located where the indicator defined by Eq.(35) is greater than auser-defined threshold. The figure 13 shows the measured phase map changewhen the cantilever is submitted to bimaterial effect. One distinguish thesubstrate, which is not subjected to any modification. One then remark thephase change is not homogeneous across the cantilever.

The displacement field is then recovered using the algorithm described inSection 44.1. The function basis is chosen to be able to represent the expectedmechanical effects. As a consequence, one chooses cubic hermite polynomials[18] by part along the y direction. In the present example, four elements alongthe cantilever are found to be sufficient to describe heterogeneous effects,and the projection is made independently for each line across the beam.The figure 14 shows the resulting displacement field, which clearly exhibitsan heterogeneous behavior across the cantilever. One should emphasize,that the free end of the cantilever experiences a phase change of 0.8 rad(Fig.13). This value is to be compared to the one arising only from a 12 nmheight modification (Fig. 14), which is 0.18 rad (first term in Eq.(51)) ; thusdemonstrating the necessity of taking both height and slope changes intoaccount. Moreover, the displacement field shows the cantilever part close tothe anchoring rises up as the temperature increases, whereas the cantilever asa whole bends down. This is thought to be the signature of an under-etchedcantilever, as the remaining silica pedestal dilates and pushes the surfaceup. This example typically illustrates the extensive amount of informationsprovided by full-field measurement compared to pointwise ones.

5 Conclusions

A Nomarski imaging interferometer is used to measure the differential topog-raphy of reflective objects. A phase modulation is introduced to measure thephase map arising from the object with a sensitivity independent on the ac-tual phase value. The phase measurement is shown to be shot-noise limited,and a measurement reproducibility of almost 10 pm is achieved. The use offour integrating buckets with a least-square algorithm is shown to improvethe phase retrieval in case of low fringe constrast. Moreover, this allows to

16

reconsider the phase unwrapping problem, so that it is easy to deal withhighly curved objects.

The physical origin of the measured optical phase is calculated, and ex-hibits that the phase difference is due to both height and slope variations.A simple calibration procedure allows one to check for the phase sensitiv-ity to slope variations. These two effects are then decoupled expanding thedisplacement field onto a chosen functions basis.

The out-of-plane displacement field of a MEMS cantilever subjected tobimaterial effect is then recovered thereby proving the ability of such a set-upto provide a reliable full-field kinematic measurement without surface mod-ification. This tool provides then a full displacement field using a common-path interferometer. Moreover, its stability, as well as its ability to operatethrough a wide range of media make this set-up a very powerful tool forthe study of the mechanical behavior of micro-electro-mechanical systems,providing an extensive amount of reliable information.

17

A Appendix : Effect of the numerical aper-

ture

The section 22.4 is devoted to the calculation of the optical phase differencearising from the topography of the object under scope. This difference de-pends on the incidence of the ray on the surface. When using an imagingsystem, this difference depends therefore on the numerical aperture of theobjective lens. The expression (18) was obtained under the assumption thatthe incidence angle is low enough. To assess this assumption’s validity, let usconsider the contribution of each incidence in the figure’s plane to the totalintensity in Eq.4

dI(α) =

(

dI0 + dA cos

[

4nπ∆z

λcos(α) + ψ

])

dα (60)

where ψ stands for the optical phase independent from the topography. Asthe fringe spacing depends on the angle α between the ray and the optic axis,the optical phase arising from the step in Fig. 5 is obtained by weightingand summing the contributions of each rays impinging on the sample at apoint of the field of view

I = I0 + A2

sin2(αmax)

∫ αmax

0

cos

[

4nπ∆z

λcos(α) + ψ

]

P (α)2 sin(α)dα (61)

αmax is related to the numerical aperture of the objective lens NA

NA = n sin(αmax) (62)

and to the apodization function P (α). The choice of this function has beenwidely discussed [19, 20, 21] and let us the assume

P (α) = (cos(α))m (63)

where m is a parameter used to describe the apodization effect. The equation(4) turns into

I = I0 + AmFNA,m(∆z, ψ) (64)

for m = 0 (Herschel condition),

FNA,0(∆z, ψ) ≃sin(k∆z(1 − cos(αmax)))

k∆z(1 − cos(αmax))cos(k∆z(1+cos(αmax))+ψ) (65)

18

where k = 2nπλ. If m 6= 0, one can derive the Taylor expansion of (64) with

respect to αmax

FNA,m(∆z, ψ) =2

sin2(αmax)

∫ αmax

0

cos(2k∆z cos(α) + ψ)P (α)2m sin(α)dα

≃ cos(2k∆z + ψ) +

(

1− 2m

4cos(2k∆z + ψ) +

1

2sin(2k∆z + ψ)k∆z

)

α2max

+

(

1 + 4(m2 −m− k2∆z2)

24cos(2k∆z + ψ)

+k∆z(1 − 4m)

12sin(2k∆z + ψ)

)

α4max + . . . (66)

Let us define the fringe spacing iinterf , as the value of k∆z where the intensity(64) reaches its first local maximum. As the numerical aperture increases,iinterf 6= π. Let us thus define the gap

ǫinterf = 1− iinterfπ

(67)

Computing ǫinterf , when the numerical aperture ranges from 0 to 0.6, and mranges from 0 to 2 shows that ǫinterf < 0 so that the fringe spacing increaseswith the numerical aperture (see equation (67)). if the numerical apertureis less than 0.4, ǫinterf no longer depends significantly on m, so that thenumerical aperture value is sufficient to retrieve ∆z from the intensity value.The above correction of the equation (18) is easy as long as FNA,m(∆z, ψ) ispseudo-periodic in a ∆z range sufficient to describe the topography. This isassessed by defining the ratio

ri =i10

10iinterf(68)

where i10 is the k∆z value for which FNA,m(∆z, ψ) reaches its tenth localmaximum If the numerical aperture is less than 0.3, ri is 1, thereby provingthat FNA,m(∆z, ψ) is pseudo-periodic on the defined range. The correctionof Eq. (18) then reads

φh =4πn∆z

ιλ(69)

where ι depends on the numerical aperture and the apodization function.The figure 15 shows the value of ι as a function of both the numerical apertureand m. If the numerical aperture is less than 0.3, ι no longer depends on m,and therefore allows for a direct retrieval of the phase φh.

19

References

[1] N.V. Lavrik, M.J. Sepaniak and P.G. Datskos, “Cantilever transducersas a platform for chemical and biological sensors,” Rev. Sci. Instrum.75, 2229-2253 (2004).

[2] T.P. Weihs, S. Hong, J.C. Bravman and W.D. Nix, “Mechanicaldeflection of cantilever micro-beams: a new technique for testing themechanical properties of thin films,” J. Mater. Res. 3, 931–942 (1988).

[3] D.T. Read and J.W. Dally, “A new method for measuring the strengthand ductility of thin films,” J. Mater. Res. 8, 1542–1549 (1993).

[4] M.A. Haque and M.T.A. Saif, “A review of MEMS-Based microscale andnanoscale tensile and bending testing,” Exp. Mech. 43, 248–255 (2003).

[5] M. Gad-El-Hak, The MEMS Handbook , (CRC Press, 2002).

[6] Y.Y. Hung and C.Y. Liang, “Image-shearing camera for direct mea-surement of surface strains,” Appl. Opt. 18, 1046–1051 (1979).

[7] P. Aswendt, C.-D. Schmidt, D. Zielke and S. Schubert, “ESPI solutionfor non-contacting MEMS-on-wafer testing,” Optics and Lasers inEngineering 40, 501–515 (2003).

[8] M. Francon, “Polarization interference microscopes,” Appl. Optics 3,

1033-1036 (1964).

[9] G. Nomarski, “Microinterferometre differentiel a ondes polarisees,” J.Phys. Radium 16, 9S–11S (1955).

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[10] D.L. Lessor, J.S. Hartman and R.L. Gordon, “Quantitative surface to-pography determination by Nomarski reflection microscopy. I. Theory,”J. Opt. Soc. Am. 69, 357–365 (1979).

[11] P. Gleyzes and A.C. Boccara,“ Profilometrie picometrique par in-terferometrie de polarisation. I. L’approche monodetecteur,” J. Optics(Paris) 25, 207–224 (1994).

[12] P. Gleyzes, F. Guernet and A.C. Boccara, “Profilometrie picometrique.II L’approche multi-detecteur et la detection synchrone multiplexee.,”J. Optics (Paris) 26, 251–265 (1995).

[13] P. Gleyzes, A.C. Boccara and H. Saint-Jalmes, “Multichannel Nomarskimicroscope with polarization modulation: performance and applica-tions”, Optics Letters 22, 1529–1531 (1997).

[14] C. Montarou and T.K. Gaylord, “Analysis and design of modifiedWollaston prisms”, Appl. Opt. 38, 6604–6616 (1999).

[15] A. Dubois, “Phase-map measurements by interferometry with sinuoidalphase modulation and four integrating buckets,” J. Opt. Soc. Am. A18, 1972-1979 (2001).

[16] R. Cusack, J.M. Huntley and H.T. Goldrein, “Improved noise-immunephase-unwrapping algorithm,” Appl. Opt. 34, 781-789 (1995).

[17] X.Y. He, X. Kang, C.J. Tay, C. Quan and H.M. Shang, “Proposedalgorithm for phase unwrapping,” Appl. Opt. 41, 7422-7428 (2002).

[18] S.P. Timoshenko and J.N. Goodier, Theory of Elasticity , (McGraw-Hill(3rd edition), New York (USA), 1970).

21

[19] A. Dubois, J. Selb, L. Vabre and A.C. Boccara, “Phase measurementswith wide-aperture interferometers,” Appl. Opt. 39, 2326-2331 (2000).

[20] C.R.J. Sheppard and K.G. Larkin, “Effect of numerical aperture oninterference fringe spacing,” Appl. Opt. 34, 4731-4734 (1995).

[21] G. Schulz and K.-E. Elssner, “Errors in phase-measurement interferom-etry with high numerical apertures,” Appl. Opt. 30, 4500-4506 (1991).

22

List of Figures

1 Schematic view of the basic interferential microscopy imaging set-up. 242 Typical interference pattern obtained in water with two 70× 20× 0.84µm3 microcantilevers and a shear distance d ≃ 50µm (NA=0.3). 253 Schematic view of a Wollaston prism. . . . . . . . . . . . . . . 264 Ray tracing for a plane object. . . . . . . . . . . . . . . . . . . 275 Ray tracing in the case of a tilted and stepped sample (height ∆z). 286 Ray tracing for a sample subjected to slope variations. . . . . 297 Calibration measurement of the mean phase induced by a sample as a function of its tilt. 308 Typical wrapped phase map obtained in water with two 70× 20× 0.84µm3 microcantilevers and a shear distance d = 53.4µm (NA=0.3). 319 Phase unwrapping principle. . . . . . . . . . . . . . . . . . . . 3210 Typical unwrapped phase map obtained in water with two 70× 20× 0.84µm3 microcantilevers and a shear distance d = 53.4µm (NA=0.3). 3311 Typical experimental phase noise probability density (dots) and its fit by a Gaussian distribution (solid line). 3412 Estimation of the reproducibility on the measurement of a differential topography as a function of the exposure time. 3513 Measured phase map change when the cantilever is subjected to bimaterial effect. 3614 Displacement field calculated from the measured phase map change shown in Fig.13. 3715 Evolution of the correction factor ι when the numerical aperture ranges from 0 to 0.4, and the exponent m ranges from 0 to 2. 38

23

Figure 1: Schematic view of the basic interferential microscopy imaging set-up.

24

Figure 2: Typical interference pattern obtained in water with two 70× 20×0.84µm3 microcantilevers and a shear distance d ≃ 50µm (NA=0.3).

25

Figure 3: Schematic view of a Wollaston prism.

26

Figure 4: Ray tracing for a plane object.

27

Figure 5: Ray tracing in the case of a tilted and stepped sample (height ∆z).

28

Figure 6: Ray tracing for a sample subjected to slope variations.

29

Figure 7: Calibration measurement of the mean phase induced by a sampleas a function of its tilt.

30

Figure 8: Typical wrapped phase map obtained in water with two 70× 20×0.84µm3 microcantilevers and a shear distance d = 53.4µm (NA=0.3).

31

Figure 9: Phase unwrapping principle.

32

Figure 10: Typical unwrapped phase map obtained in water with two 70 ×20× 0.84µm3 microcantilevers and a shear distance d = 53.4µm (NA=0.3).

33

Figure 11: Typical experimental phase noise probability density (dots) andits fit by a Gaussian distribution (solid line).

34

Figure 12: Estimation of the reproducibility on the measurement of a differ-ential topography as a function of the exposure time.

35

Figure 13: Measured phase map change when the cantilever is subjected tobimaterial effect.

36

Figure 14: Displacement field calculated from the measured phase mapchange shown in Fig.13.

37

Figure 15: Evolution of the correction factor ι when the numerical apertureranges from 0 to 0.4, and the exponent m ranges from 0 to 2.

38